Inertial range scaling in numerical turbulence with hyperviscosity

Evolution of weak, homogeneous turbulence subject to rotation and stratification: Comparable wave and nonpropagating components

Following on from previous work [J. F. Scott and C. Cambon, J. Fluid Mech. 979, A17 (2024)10.1017/jfm.2023.1046], this article concerns weak (small Rossby or Froude number), homogeneous turbulence subject to rotation and stable stratification. The flow is expressed as a combination of particular solutions (modes) of the linearized governing equations without viscosity or diffusion. Modes are of two types: oscillatory ones which represent inertial-gravity waves and time-independent ones that express a nonpropagating (NP) component of the flow. It was shown in the previous work that, at leading order, the NP component evolves independently of the wave component and a specifically adapted direct numerical simulation (DNS) approach was introduced to describe the NP component, the same approach which is employed here. Using wave-turbulence analysis and assuming the NP amplitude is small compared with the wave amplitude, evolution equations for the wave spectra were derived and numerically exploited. Here, those equations are extended to the case when the NP and wave components are of comparable magnitude. The NP spectra then appear in the wave-turbulence equations, which means those equations are no longer closed. As a result, a combination of adapted DNS for the NP component and the wave-turbulence equations is used and numerical solutions of the latter are given. Terms in the wave-turbulence equations arising from the NP component couple pairs of wave modes, adding to the three-wave interactions of the previous work when the latter exist. This is found to considerably increase the wave dissipation. Indeed, it provides the only mechanism for significant dissipation in the cases for which three-wave interactions are absent. The additional dissipation is especially important for wave modes having wave vectors perpendicular to the vertical or rotation axis, but is also effective for other directions.

Statistics of relative velocity for particles settling under gravity in a turbulent flow

We study the joint probability distributions of separation R and radial component of the relative velocity V_{R} of particles settling under gravity in a turbulent flow. We also obtain the moments of these distributions and analyze their anisotropy using spherical harmonics. We find that the qualitative nature of the joint distributions remains the same as no-gravity case. Distributions of V_{R} for fixed values of R show a power-law dependence on V_{R} for a range of V_{R}; the exponent of the power law depends on the gravity. Effects of gravity are also manifested in the following ways: (a) Moments of the distributions are anisotropic; degree of anisotropy depends on particle's Stokes number, but does not depend on R for small values of R. (b) Mean velocity of collision between two particles is decreased for particles having equal Stokes numbers but increased for particles having different Stokes numbers. For the later, collision velocity is set by the difference in their settling velocities.

Statistics of the relative velocity of particles in turbulent flows: Monodisperse particles

We use direct numerical simulations to calculate the joint probability density function of the relative distance R and relative radial velocity component V_{R} for a pair of heavy inertial particles suspended in homogeneous and isotropic turbulent flows. At small scales the distribution is scale invariant, with a scaling exponent that is related to the particle-particle correlation dimension in phase space, D_{2}. It was argued [K. Gustavsson and B. Mehlig, Phys. Rev. E 84, 045304 (2011)PLEEE81539-375510.1103/PhysRevE.84.045304; J. Turbul. 15, 34 (2014)1468-524810.1080/14685248.2013.875188] that the scale invariant part of the distribution has two asymptotic regimes: (1) |V_{R}|≪R, where the distribution depends solely on R, and (2) |V_{R}|≫R, where the distribution is a function of |V_{R}| alone. The probability distributions in these two regimes are matched along a straight line: |V_{R}|=z^{*}R. Our simulations confirm that this is indeed correct. We further obtain D_{2} and z^{*} as a function of the Stokes number, St. The former depends nonmonotonically on St with a minimum at about St≈0.7 and the latter has only a weak dependence on St.

Wavelet-based regularization of the Galerkin truncated three-dimensional incompressible Euler flows

We present numerical simulations of the three-dimensional Galerkin truncated incompressible Euler equations that we integrate in time while regularizing the solution by applying a wavelet-based denoising. For this, at each time step, the vorticity field is decomposed into wavelet coefficients, which are split into strong and weak coefficients, before reconstructing them in physical space to obtain the corresponding coherent and incoherent vorticities. Both components are multiscale and orthogonal to each other. Then, by using the Biot-Savart kernel, one obtains the coherent and incoherent velocities. Advancing the coherent flow in time, while filtering out the noiselike incoherent flow, models turbulent dissipation and corresponds to an adaptive regularization. To track the flow evolution in both space and scale, a safety zone is added in wavelet coefficient space to the coherent wavelet coefficients. It is shown that the coherent flow indeed exhibits an intermittent nonlinear dynamics and a k^{-5/3} energy spectrum, where k is the wave number, characteristic of three-dimensional homogeneous isotropic turbulence. Finally, we compare the dynamical and statistical properties of Euler flows subjected to four kinds of regularizations: dissipative (Navier-Stokes), hyperdissipative (iterated Laplacian), dispersive (Euler-Voigt), and wavelet-based regularizations.

Nonhelical turbulence and the inverse transfer of energy: A parameter study

We explore the phenomenon of the recently discovered inverse transfer of energy from small to large scales in decaying magnetohydrodynamical turbulence by Brandenburg et al. [Phys. Rev. Lett. 114, 075001 (2015)PRLTAO0031-900710.1103/PhysRevLett.114.075001], even for nonhelical magnetic fields. For this investigation we mainly employ the Pencil Code performing a parameter study, where we vary the Prandtl number, the kinematic viscosity, and the initial spectrum. We find that to get a decay that exhibits this inverse transfer, large Reynolds numbers (O∼10^{3}) are needed and low Prandtl numbers of the order unity Pr=1 are preferred. Compared to helical MHD turbulence, though, the inverse transfer is much less efficient in transferring magnetic energy to larger scales than the well-known effect of the inverse cascade. Hence, applying the inverse transfer to the magnetic field evolution in the Early Universe, we question whether the nonhelical inverse transfer is effective enough to explain the observed void magnetic fields if a magnetogenesis scenario during the electroweak phase transition is assumed.

Classes of Hydrodynamic and Magnetohydrodynamic Turbulent Decay

We perform numerical simulations of decaying hydrodynamic and magnetohydrodynamic turbulence. We classify our time-dependent solutions by their evolutionary tracks in parametric plots between instantaneous scaling exponents. We find distinct classes of solutions evolving along specific trajectories toward points on a line of self-similar solutions. These trajectories are determined by the underlying physics governing individual cases, while the infrared slope of the initial conditions plays only a limited role. In the helical case, even for a scale-invariant initial spectrum (inversely proportional to wave number k), the solution evolves along the same trajectory as for a Batchelor spectrum (proportional to k^{4}).

Enhancement of intermittency in superfluid turbulence

We consider the intermittent behavior of superfluid turbulence in (4)He. Because of the similarity in the nonlinear structure of the two-fluid model of superfluidity and the Euler and Navier-Stokes equations, one expects the scaling exponents of the structure functions to be the same as in classical turbulence for temperatures close to the superfluid transition T(λ) and also for T << T(λ). This is not the case when the densities of normal and superfluid components are comparable to each other and mutual friction becomes important. Using shell model simulations, we propose that in this situation there exists a range of scales in which the effective exponents indicate stronger intermittency. We offer a bridge relation between these effective and the classical scaling exponents. Since this effect occurs at accessible temperatures and Reynolds numbers, we propose that experiments should be conducted to further assess the validity and implications of this prediction.

Astrophysical turbulence modeling

The role of turbulence in various astrophysical settings is reviewed. Among the differences to laboratory and atmospheric turbulence we highlight the ubiquitous presence of magnetic fields that are generally produced and maintained by dynamo action. The extreme temperature and density contrasts and stratifications are emphasized in connection with turbulence in the interstellar medium and in stars with outer convection zones, respectively. In many cases turbulence plays an essential role in facilitating enhanced transport of mass, momentum, energy and magnetic fields in terms of the corresponding coarse-grained mean fields. Those transport properties are usually strongly modified by anisotropies and often completely new effects emerge in such a description that have no correspondence in terms of the original (non-coarse-grained) fields.

Hyperviscosity, Galerkin truncation, and bottlenecks in turbulence

It is shown that the use of a high power alpha of the Laplacian in the dissipative term of hydrodynamical equations leads asymptotically to truncated inviscid conservative dynamics with a finite range of spatial Fourier modes. Those at large wave numbers thermalize, whereas modes at small wave numbers obey ordinary viscous dynamics [C. Cichowlas et al., Phys. Rev. Lett. 95, 264502 (2005)10.1103/Phys. Rev. Lett. 95.264502]. The energy bottleneck observed for finite alpha may be interpreted as incomplete thermalization. Artifacts arising from models with alpha>1 are discussed.

Compressibility in solar wind plasma turbulence

Incompressible magnetohydrodynamics is often assumed to describe solar wind turbulence. We use extended self-similarity to reveal scaling in the structure functions of density fluctuations in the solar wind. The obtained scaling is then compared with that found in the inertial range of quantities identified as passive scalars in other turbulent systems. We find that these are not coincident. This implies that either solar wind turbulence is compressible or that straightforward comparison of structure functions does not adequately capture its inertial range properties.

Delayed correlation between turbulent energy injection and dissipation

The dimensionless kinetic energy dissipation rate C(epsilon) is estimated from numerical simulations of statistically stationary isotropic box turbulence that is slightly compressible. The Taylor microscale Reynolds number (Re(lambda)) range is 20< or approximately equal to Re(lambda) < or approximately equal to 220 and the statistical stationarity is achieved with a random phase forcing method. The strong Re(lambda) dependence of C(epsilon) abates when Re(lambda) approximately 100 after which C(epsilon) slowly approaches approximately 0.5, a value slightly different from previously reported simulations but in good agreement with experimental results. If C(epsilon) is estimated at a specific time step from the time series of the quantities involved it is necessary to account for the time lag between energy injection and energy dissipation. Also, the resulting value can differ from the ensemble averaged value by up to +/-30%. This may explain the spread in results from previously published estimates of C(epsilon).